Group #28297971 — subgroups

Subgroups — C₂⁴×Q₈×D₅₃ #28297971

Order 13568.

63326 conjugacy classes of subgroups; only the 34795 normal subgroups are shown.

normal subgroups only

This group has 63326 conjugacy classes of subgroups — too many to draw. Only its 34795 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×Dic₁₀₆	C₂	1	3	yes	maximal, complemented
2	C₂³×Q₈×D₅₃	C₂	1	120	yes	maximal, direct product factor
3	C₂⁴×C₄×D₅₃	C₂	1	3	yes	maximal
4	C₂⁴×Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂⁵×D₅₃	C₂²	1	1	yes	characteristic
6	C₂³×Dic₁₀₆	C₂²	1	180	yes	complemented
7	C₂²×Q₈×D₅₃	C₂²	1	2240	yes	direct product factor
8	C₂⁴×Dic₅₃	C₂²	1	3	yes	—
9	C₂³×C₄×D₅₃	C₂²	1	180	yes	—
10	C₂³×Q₈×C₅₃	C₂²	1	60	yes	complemented
11	C₂⁴×C₄×C₅₃	C₂²	1	3	yes	—
12	C₂⁴×D₅₃	Q₈	1	32	yes	direct product factor
13	C₂⁴×D₅₃	C₂³	1	30	yes	—
14	C₂²×Dic₁₀₆	C₂³	1	1680	yes	complemented
15	C₂×Q₈×D₅₃	C₂³	1	7680	yes	direct product factor
16	C₂³×Dic₅₃	C₂³	1	90	yes	—
17	C₂²×C₄×D₅₃	C₂³	1	1680	yes	—
18	C₂⁵×C₅₃	C₂³	1	1	yes	characteristic
19	C₂²×Q₈×C₅₃	C₂³	1	560	yes	complemented
20	C₂³×C₄×C₅₃	C₂³	1	90	yes	—
21	C₂⁴×C₅₃	C₂×Q₈	1	16	yes	complemented
22	C₂⁴×C₅₃	C₂⁴	1	15	yes	—
23	C₂³×D₅₃	C₂×Q₈	1	480	yes	direct product factor
24	C₂³×D₅₃	C₂⁴	1	140	yes	—
25	C₂×Q₈×C₅₃	C₂⁴	1	960	yes	complemented
26	C₂²×C₄×C₅₃	C₂⁴	1	420	yes	—
27	C₂²×Dic₅₃	C₂⁴	1	420	yes	—
28	Q₈×D₅₃	C₂⁴	1	4096	yes	direct product factor
29	C₂×C₄×D₅₃	C₂⁴	1	2880	yes	—
30	C₂×Dic₁₀₆	C₂⁴	1	2880	yes	complemented
31	C₂³×C₅₃	C₂²×Q₈	1	120	yes	complemented
32	C₂³×C₅₃	C₂⁵	1	35	yes	—
33	C₂²×D₅₃	C₂²×Q₈	1	1120	yes	direct product factor
34	C₂²×D₅₃	C₂⁵	1	120	yes	—
35	Q₈×C₅₃	C₂⁵	1	256	yes	complemented
36	C₂×C₄×C₅₃	C₂⁵	1	360	yes	—
37	C₂×Dic₅₃	C₂⁵	1	360	yes	—
38	C₄×D₅₃	C₂⁵	1	768	yes	—
39	Dic₁₀₆	C₂⁵	1	768	yes	complemented
40	C₂⁵×Q₈	—	53	1	no	maximal, Sylow
41	C₂²×C₅₃	C₂³×Q₈	1	140	yes	complemented
42	C₂²×C₅₃	C₂⁶	1	15	yes	—
43	C₂×D₅₃	C₂³×Q₈	1	480	yes	direct product factor
44	C₂×D₅₃	C₂⁶	1	16	yes	—
45	C₄×C₅₃	C₂⁶	1	48	yes	—
46	Dic₅₃	C₂⁶	1	48	yes	—
47	C₂⁴×Q₈	—	53	120	no	—
48	C₂⁴×Q₈	—	53	3	no	—
49	C₂⁴×Q₈	D₅₃	1	1	yes	characteristic, direct product factor
50	C₂⁵×C₄	—	53	3	no	—
51	C₂×C₅₃	C₂⁴×Q₈	1	30	yes	complemented
52	C₂×C₅₃	C₂⁷	1	1	yes	characteristic
53	D₅₃	C₂⁴×Q₈	1	32	yes	direct product factor
54	C₂⁶	—	53	1	no	—
55	C₂³×Q₈	—	53	2240	no	—
56	C₂³×Q₈	—	53	180	no	—
57	C₂³×Q₈	C₂×D₅₃	1	60	yes	direct product factor
58	C₂⁴×C₄	—	53	3	no	—
59	C₂⁴×C₄	—	53	180	no	—
60	C₂⁴×C₄	C₂×D₅₃	1	3	yes	—
61	C₅₃	C₂⁵×Q₈	1	1	yes	characteristic, complemented, Sylow
62	C₂⁵	—	53	32	no	—
63	C₂⁵	—	53	30	no	—
64	C₂⁵	C₂²×D₅₃	1	1	yes	characteristic
65	C₂²×Q₈	—	53	7680	no	—
66	C₂²×Q₈	—	53	1680	no	—
67	C₂²×Q₈	C₂²×D₅₃	1	560	yes	direct product factor
68	C₂³×C₄	—	53	90	no	—
69	C₂³×C₄	—	53	1680	no	—
70	C₂³×C₄	C₂²×D₅₃	1	90	yes	—
71	C₂⁴	—	53	480	no	—
72	C₂⁴	—	53	140	no	—
73	C₂⁴	Q₈×D₅₃	1	16	yes	direct product factor
74	C₂⁴	C₂³×D₅₃	1	15	yes	—
75	C₂×Q₈	—	53	4096	no	—
76	C₂×Q₈	—	53	2880	no	—
77	C₂×Q₈	C₂³×D₅₃	1	960	yes	direct product factor
78	C₂²×C₄	—	53	420	no	—
79	C₂²×C₄	—	53	2880	no	—
80	C₂²×C₄	C₂³×D₅₃	1	420	yes	—
81	C₂³	—	53	1120	no	—
82	C₂³	—	53	120	no	—
83	C₂³	C₂×Q₈×D₅₃	1	120	yes	direct product factor
84	C₂³	C₂⁴×D₅₃	1	35	yes	—
85	Q₈	—	53	768	no	—
86	Q₈	C₂⁴×D₅₃	1	256	yes	direct product factor
87	C₂×C₄	—	53	360	no	—
88	C₂×C₄	—	53	768	no	—
89	C₂×C₄	C₂⁴×D₅₃	1	360	yes	—
90	C₂²	—	53	480	no	—
91	C₂²	—	53	16	no	—
92	C₂²	C₂²×Q₈×D₅₃	1	140	yes	direct product factor
93	C₂²	C₂⁵×D₅₃	1	15	yes	—
94	C₄	—	53	48	no	—
95	C₄	C₂⁵×D₅₃	1	48	yes	—
96	C₂	—	53	32	no	—
97	C₂	C₂³×Q₈×D₅₃	1	30	yes	direct product factor
98	C₂	C₂⁶×D₅₃	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂⁴×Dic₁₀₆

C₂

yes

maximal, complemented

C₂³×Q₈×D₅₃

C₂

120

yes

maximal, direct product factor

C₂⁴×C₄×D₅₃

C₂

yes

maximal

C₂⁴×Q₈×C₅₃

C₂

yes

maximal, characteristic, complemented

C₂⁵×D₅₃

C₂²

yes

characteristic

C₂³×Dic₁₀₆

C₂²

180

yes

complemented

C₂²×Q₈×D₅₃

C₂²

2240

yes

direct product factor

C₂⁴×Dic₅₃

C₂²

yes

—

C₂³×C₄×D₅₃

C₂²

180

yes

—

C₂³×Q₈×C₅₃

C₂²

yes

complemented

C₂⁴×C₄×C₅₃

C₂²

yes

—

C₂⁴×D₅₃

Q₈

yes

direct product factor

C₂⁴×D₅₃

C₂³

yes

—

C₂²×Dic₁₀₆

C₂³

1680

yes

complemented

C₂×Q₈×D₅₃

C₂³

7680

yes

direct product factor

C₂³×Dic₅₃

C₂³

yes

—

C₂²×C₄×D₅₃

C₂³

1680

yes

—

C₂⁵×C₅₃

C₂³

yes

characteristic

C₂²×Q₈×C₅₃

C₂³

560

yes

complemented

C₂³×C₄×C₅₃

C₂³

yes

—

C₂⁴×C₅₃

C₂×Q₈

yes

complemented

C₂⁴×C₅₃

C₂⁴

yes

—

C₂³×D₅₃

C₂×Q₈

480

yes

direct product factor

C₂³×D₅₃

C₂⁴

140

yes

—

C₂×Q₈×C₅₃

C₂⁴

960

yes

complemented

C₂²×C₄×C₅₃

C₂⁴

420

yes

—

C₂²×Dic₅₃

C₂⁴

420

yes

—

Q₈×D₅₃

C₂⁴

4096

yes

direct product factor

C₂×C₄×D₅₃

C₂⁴

2880

yes

—

C₂×Dic₁₀₆

C₂⁴

2880

yes

complemented

C₂³×C₅₃

C₂²×Q₈

120

yes

complemented

C₂³×C₅₃

C₂⁵

yes

—

C₂²×D₅₃

C₂²×Q₈

1120

yes

direct product factor

C₂²×D₅₃

C₂⁵

120

yes

—

Q₈×C₅₃

C₂⁵

256

yes

complemented

C₂×C₄×C₅₃

C₂⁵

360

yes

—

C₂×Dic₅₃

C₂⁵

360

yes

—

C₄×D₅₃

C₂⁵

768

yes

—

Dic₁₀₆

C₂⁵

768

yes

complemented

C₂⁵×Q₈

—

maximal, Sylow

C₂²×C₅₃

C₂³×Q₈

140

yes

complemented

C₂²×C₅₃

C₂⁶

yes

—

C₂×D₅₃

C₂³×Q₈

480

yes

direct product factor

C₂×D₅₃

C₂⁶

yes

—

C₄×C₅₃

C₂⁶

yes

—

Dic₅₃

C₂⁶

yes

—

C₂⁴×Q₈

—

120

—

C₂⁴×Q₈

—

C₂⁴×Q₈

D₅₃

yes

characteristic, direct product factor

C₂⁵×C₄

—

C₂×C₅₃

C₂⁴×Q₈

yes

complemented

C₂×C₅₃

C₂⁷

yes

characteristic

D₅₃

C₂⁴×Q₈

yes

direct product factor

C₂⁶

—

C₂³×Q₈

—

2240

—

C₂³×Q₈

—

180

—

C₂³×Q₈

C₂×D₅₃

yes

direct product factor

C₂⁴×C₄

—

C₂⁴×C₄

—

180

—

C₂⁴×C₄

C₂×D₅₃

yes

—

C₅₃

C₂⁵×Q₈

yes

characteristic, complemented, Sylow

C₂⁵

—

C₂⁵

—

C₂⁵

C₂²×D₅₃

yes

characteristic

C₂²×Q₈

—

7680

—

C₂²×Q₈

—

1680

—

C₂²×Q₈

C₂²×D₅₃

560

yes

direct product factor

C₂³×C₄

—

C₂³×C₄

—

1680

—

C₂³×C₄

C₂²×D₅₃

yes

—

C₂⁴

—

480

—

C₂⁴

—

140

—

C₂⁴

Q₈×D₅₃

yes

direct product factor

C₂⁴

C₂³×D₅₃

yes

—

C₂×Q₈

—

4096

—

C₂×Q₈

—

2880

—

C₂×Q₈

C₂³×D₅₃

960

yes

direct product factor

C₂²×C₄

—

420

—

C₂²×C₄

—

2880

—

C₂²×C₄

C₂³×D₅₃

420

yes

—

C₂³

—

1120

—

C₂³

—

120

—

C₂³

C₂×Q₈×D₅₃

120

yes

direct product factor

C₂³

C₂⁴×D₅₃

yes

—

Q₈

—

768

—

Q₈

C₂⁴×D₅₃

256

yes

direct product factor

C₂×C₄

—

360

—

C₂×C₄

—

768

—

C₂×C₄

C₂⁴×D₅₃

360

yes

—

C₂²

—

480

—

C₂²

—

C₂²

C₂²×Q₈×D₅₃

140

yes

direct product factor

C₂²

C₂⁵×D₅₃

yes

—

C₄

—

C₄

C₂⁵×D₅₃

yes

—

C₂

—

C₂

C₂³×Q₈×D₅₃

yes

direct product factor

C₂

C₂⁶×D₅₃

yes

characteristic

	subgroup	G/H
center	C₂⁵	C₂²×D₅₃
commutator, cosocle	C₂×C₅₃	C₂⁷
Frattini	C₂	C₂⁶×D₅₃
Fitting	C₂⁴×Q₈×C₅₃	C₂
socle	C₂⁵×C₅₃	C₂³
prime radical	C₂⁵×D₅₃	C₂²
hypercenter	C₂⁴×Q₈	D₅₃
hypocenter	C₅₃	C₂⁵×Q₈
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂⁵ ◁ C₂⁴×Q₈

subgroup

G/H

center

C₂⁵

C₂²×D₅₃

commutator, cosocle

C₂×C₅₃

C₂⁷

Frattini

C₂

C₂⁶×D₅₃

Fitting

C₂⁴×Q₈×C₅₃

C₂

socle

C₂⁵×C₅₃

C₂³

prime radical

C₂⁵×D₅₃

C₂²

hypercenter

C₂⁴×Q₈

D₅₃

hypocenter

C₅₃

C₂⁵×Q₈

LCS

G ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂⁵ ◁ C₂⁴×Q₈

Complements

N	H ≅ G/N	type
1. C₂⁴×Dic₁₀₆	C₂	semidirect
2. C₂³×Q₈×D₅₃	C₂	direct (N × H)
4. C₂⁴×Q₈×C₅₃	C₂	semidirect
6. C₂³×Dic₁₀₆	C₂²	semidirect
7. C₂²×Q₈×D₅₃	C₂²	direct (N × H)
10. C₂³×Q₈×C₅₃	C₂²	semidirect
12. C₂⁴×D₅₃	Q₈	direct (N × H)
14. C₂²×Dic₁₀₆	C₂³	semidirect
15. C₂×Q₈×D₅₃	C₂³	direct (N × H)
19. C₂²×Q₈×C₅₃	C₂³	semidirect
21. C₂⁴×C₅₃	C₂×Q₈	semidirect
23. C₂³×D₅₃	C₂×Q₈	direct (N × H)
25. C₂×Q₈×C₅₃	C₂⁴	semidirect
28. Q₈×D₅₃	C₂⁴	direct (N × H)
30. C₂×Dic₁₀₆	C₂⁴	semidirect
31. C₂³×C₅₃	C₂²×Q₈	semidirect
33. C₂²×D₅₃	C₂²×Q₈	direct (N × H)
35. Q₈×C₅₃	C₂⁵	semidirect
39. Dic₁₀₆	C₂⁵	semidirect
41. C₂²×C₅₃	C₂³×Q₈	semidirect
43. C₂×D₅₃	C₂³×Q₈	direct (N × H)
49. C₂⁴×Q₈	D₅₃	direct (N × H)
51. C₂×C₅₃	C₂⁴×Q₈	semidirect
53. D₅₃	C₂⁴×Q₈	direct (N × H)
57. C₂³×Q₈	C₂×D₅₃	direct (N × H)
61. C₅₃	C₂⁵×Q₈	semidirect
67. C₂²×Q₈	C₂²×D₅₃	direct (N × H)
73. C₂⁴	Q₈×D₅₃	direct (N × H)
77. C₂×Q₈	C₂³×D₅₃	direct (N × H)
83. C₂³	C₂×Q₈×D₅₃	direct (N × H)
86. Q₈	C₂⁴×D₅₃	direct (N × H)
92. C₂²	C₂²×Q₈×D₅₃	direct (N × H)
97. C₂	C₂³×Q₈×D₅₃	direct (N × H)

H ≅ G/N

type

1. C₂⁴×Dic₁₀₆

C₂

semidirect

2. C₂³×Q₈×D₅₃

C₂

direct (N × H)

4. C₂⁴×Q₈×C₅₃

C₂

semidirect

6. C₂³×Dic₁₀₆

C₂²

semidirect

7. C₂²×Q₈×D₅₃

C₂²

direct (N × H)

10. C₂³×Q₈×C₅₃

C₂²

semidirect

12. C₂⁴×D₅₃

Q₈

direct (N × H)

14. C₂²×Dic₁₀₆

C₂³

semidirect

15. C₂×Q₈×D₅₃

C₂³

direct (N × H)

19. C₂²×Q₈×C₅₃

C₂³

semidirect

21. C₂⁴×C₅₃

C₂×Q₈

semidirect

23. C₂³×D₅₃

C₂×Q₈

direct (N × H)

25. C₂×Q₈×C₅₃

C₂⁴

semidirect

28. Q₈×D₅₃

C₂⁴

direct (N × H)

30. C₂×Dic₁₀₆

C₂⁴

semidirect

31. C₂³×C₅₃

C₂²×Q₈

semidirect

33. C₂²×D₅₃

C₂²×Q₈

direct (N × H)

35. Q₈×C₅₃

C₂⁵

semidirect

39. Dic₁₀₆

C₂⁵

semidirect

41. C₂²×C₅₃

C₂³×Q₈

semidirect

43. C₂×D₅₃

C₂³×Q₈

direct (N × H)

49. C₂⁴×Q₈

D₅₃

direct (N × H)

51. C₂×C₅₃

C₂⁴×Q₈

semidirect

53. D₅₃

C₂⁴×Q₈

direct (N × H)

57. C₂³×Q₈

C₂×D₅₃

direct (N × H)

61. C₅₃

C₂⁵×Q₈

semidirect

67. C₂²×Q₈

C₂²×D₅₃

direct (N × H)

73. C₂⁴

Q₈×D₅₃

direct (N × H)

77. C₂×Q₈

C₂³×D₅₃

direct (N × H)

83. C₂³

C₂×Q₈×D₅₃

direct (N × H)

86. Q₈

C₂⁴×D₅₃

direct (N × H)

92. C₂²

C₂²×Q₈×D₅₃

direct (N × H)

97. C₂

C₂³×Q₈×D₅₃

direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂⁴×Dic₁₀₆	2
C₂³×Q₈×D₅₃	2
C₂⁴×C₄×D₅₃	2
C₂⁴×Q₈×C₅₃	2
C₂⁵×Q₈	53

subgroup

index

C₂⁴×Dic₁₀₆

C₂³×Q₈×D₅₃

C₂⁴×C₄×D₅₃

C₂⁴×Q₈×C₅₃

C₂⁵×Q₈

Subgroups — C24×Q8×D53 #28297971

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂⁴×Q₈×D₅₃ #28297971